Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy


Releases Archive. Вестник ТюмГУ. Физико-математические науки. Информатика (№7, 2013)

Nonlinear wave modelling on the surface of a two-phase mixture

About the authors:

Barinov V. A, Associate Professor, Mathematical Modeling Department, Cand. Phys. and Math. Sci., Tyumen State University
Konstantin Yu. Basinsky, Cand. Sci. (Phys.-Math.), Associate Professor, Department of Fundamental Mathematics and Mechanics, University of Tyumen; k.y.basinskij@utmn.ru


The paper discusses a nonlinear boundary value problem of wave propagation on the free surface of a two-phase (disperse) mixture. To solve the problem, a method of time-variant frequency is applied, which was devised for the solution of nonlinear wave problems with dissipation. A solution has been found to a third order approximation using a small-amplitude parameter. Nonlinear wave frequency and near-surface Stokes flow rate have been defined. They are time functions depending on the wave length and elevation (as in the classical case) and on the dispersed phase concentration. Nonlinear paths of carrier and dispersed phase particles have been found. Basing on the obtained expression for the dispersed particles paths, the nonlinear effect has been described: in wave motion dispersed particles that are heavier than carrier phase particles float, while lighter particles sink.


1. Aleshkov, Ju.Z. Diffusion of Impurities in the Ocean. Vestnik Sankt-Peterburgskogo universiteta — Saint Petersburg University Herald. 2001. Series 1. Issue. 1. Pp. 12-20.

2. Nudner, I.S., Maksimov, V.V., Majorov, Ju.B., Babchik, D.V. Modelling of Pump Dynamics in the Coastal Area. Vychislitel'nye tehnologii — Computational Technologies. 2008. Vol. 13, № S2. Pp. 60-77.

3. Barinov, V.A., Butakova, N.N. A Nonlinear Problem of Surface Waves on a Two-Phase Mixture. Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki — Computational Mathematics and Mathematical Physics. 2003. Vol. 43, №. 12. Pp. 1870-1883.

4. Barinov, V.A., Butakova, N.N. Propagation of Waves on the Free Surface of a TwoPhase Mixture. Izvestija RAN — Bulletin of the Russian Academy of Sciences. 2003. №6. Pp. 94-102.

5. Barinov, V.A., Butakova, N.N. Waves on the Free Surface of a Two-Phase Medium. Prikladnaja mehanika i tehnicheskaja fizika — Journal of Applied Mechanics and Technical Physics. 2002. Vol. 43, № 4. Pp. 27-35.

6. Aleshkov, Ju.Z., Barinov, V.A., Butakova, N.N. Nonlinear Surface Waves on a Layer of a Two-Phase Medium. Vestnik Sankt-Peterburgskogo universiteta — Saint Petersburg University Herald. 2003. Series 1. Issue 4. Pp. 64-75.

7. Sretenskij, L.N. Teorija volnovyh dvizhenij zhidkosti [Theory of Wave Motion in Fluid]. М.: Nauka, 1977. 816 p.

8. Aleshkov, Ju.Z. Teorija voln na poverhnosti tjazheloj zhidkosti [Wave Theory for Heavy Liquid Surface]. L.: Leningrad University Publ., 1981. 196 p.

9. Barinov, V.A., Basinskij, K.Ju. Nonlinear Stokes Waves on the Surface of Low-Viscosity Fluid. Vestnik Udmurtskogo universiteta — Udmurt University Herald. 2011. № 2. Pp. 112-122.

10. Barinov, V.A., Basinskij, K.Ju. The Solution of a Nonlinear Problem of Waves on the Surface Weakly-Viscous Fluid. Vestnik Sankt-Peterburgskogo universiteta — Saint Petersburg University Herald. 2011. Series 10. Issue. 2. Pp. 9-16.